Analytical and simulation studies of pedestrian flow at a crossing with random update rule
Zhong-Jun Ding, Shao-Long Yu, Kongjin Zhu, Jian-Xun Ding, Bokui Chen, Qin Shi, Rui Jiang, Bing-Hong Wang
aa r X i v : . [ n li n . C G ] M a r Analytical and simulation studies of pedestrian flow at a crossing with random update rule
Zhong-Jun Ding a , b , ∗ Shao-Long Yu a , Kongjin Zhu a , Jian-XunDing a , † Bokui Chen c , Qin Shi a , Rui Jiang b , and Bing-Hong Wang d a School of Automotive and Transportation Engineering,Hefei University of Technology, Hefei 230009, People’s Republic of China b MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology,Beijing Jiaotong University, Beijing 100044, People’s Republic of China c School of Computing, National University of Singapore, 117417, Singapore and d Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China (Dated: September 7, 2018)The intersecting pedestrian flow on the 2D lattice with random update rule is studied. Each pedestrian hasthree moving directions without the back step. Under periodic boundary conditions, an intermediate phase hasbeen found at which some pedestrians could move along the border of jamming stripes. We have performedmean field analysis for the moving and intermediate phase respectively. The analytical results agree with thesimulation results well. The empty site moves along the interface of jamming stripes when the system only hasone empty site. The average movement of empty site in one Monte Carlo step (MCS) has been analyzed throughthe master equation. Under open boundary conditions, the system exhibits moving and jamming phases. Thecritical injection probability α c shows nontrivially against the forward moving probability q . The analyticalresults of average velocity, the density and the flow rate against the injection probability in the moving phasealso agree with simulation results well.Keywords:Pedestrian flow; Monte Carlo simulations; Intermediate phase; Mean field analysis. PACS numbers: 89.40.Bb, 45.70.Vn, 64.60.My
I. INTRODUCTION
Since serious trampling accidents always happen, pedes-trian flow attracts more and more attention in recent years[1, 2]. Understanding the properties of pedestrian flow is im-portant for the design of urban facilities, traffic managementand ensuring people’s safety. Pedestrian dynamic has beenstudied in various fields including physics, engineering andmathematics. Many basic and interesting phenomena such asjamming, clogging, and lane formation have been observed[1–4].There are two main approaches for the study of pedestriandynamics [1]. The first one is designing experiment or ob-serving the real scenario through video [5–10]. The other oneis to describe the pedestrian flow by developing the delicatemodels. These models include macroscopic and microscopicones. These macroscopic models are related to the traditionaltheory of fluid mechanics, etc [1, 11]. Henderson had com-pared measurements of pedestrian flows with Navier-Stokesequations [12].Microscopic models include social force, optimal velocity,cellular automata, lattice gas model, etc. In some of thesemodels such as the social force and optimal velocity models,continuous time and space have been adopted [1, 3, 8, 13–15].While some others are placed in a discretized time and space,such as cellular automata and lattice gas models [16–22].The intersecting pedestrian flows are complex because ofconflicts between two flows with different directions. A num- ∗ Electronic address: [email protected] † Electronic address: [email protected] ber of field experiments on real intersecting pedestrian flowswith four different angles have been conducted by Guo et al.[23]. At the same time, a semi-continuous model has beendeveloped and calibrated using sample data. Lian et al. [24]have conducted a series of controlled experiments of a four-directional intersecting pedestrian flow. The average local ve-locity at high densities in the cross area is a bit larger thanthe previous study. Muramatsu et al. have investigated thejamming transitions of pedestrian flow at a crossing under theperiodic [16] and open boundary conditions [17] by the latticegas model, respectively. Hilhorst et al. have studied a latticemodel of pedestrian traffic on two crossing one-way streets[25]. Its dynamics employs the frozen shuffle update. Cividiniet al. have explained stripe formation instability and revealedthat the diagonal pattern actually consists of chevrons ratherthan straight diagonals [26, 27].The perpendicular traffic flow on two-dimensional latticehas been investigated by Biham et al. (BML ) using the cel-lular automaton model [28]. Except the moving and jammingphase, D’Souza have found an intermediate stable phase withfree-flowing regions intersecting at jammed wave fronts in theoriginal BML model [29]. Ding et al. have studied an stochas-tic BML model with random update rule (BML-R) [30]. Aphase separation phenomenon has been observed when theslow-to-start effect in the BML model is considered [31].Almost all of the models presented above are random-sequential, sublattice-parallel or parallel while the model withrandom update procedures is scarce. Since the pedestrians al-ways behave randomly in real life, the model with randomupdate procedures is considered in this paper. This paper in-vestigate effects of the random update procedures on the prop-erties of the stationary state of intersecting pedestrian flows.An intermediate phase where some pedestrians move alongthe border of jamming stripes was found. The average ve-locity of the moving and intermediate phase have been ana-lyzed through the mean filed analysis. The analytical resultsare in good agreement with the simulation ones. The emptysite moves along the interface of jamming stripes when thesystem only has one empty site. The average movement ofthe empty site in one MCS was analyzed through the mas-ter equation. The critical injection probability α c under openboundary conditions shows nontrivially against the forwardmoving probability q .This paper is organized as follows: The models are intro-duced in Section 2. In Section 3, we compare the analyticalresults with simulation ones under periodic and open bound-ary conditions, in 3.1 and 3.2, respectively. Section 4 givesthe conclusions. II. MODEL
There are two species of pedestrians distributed randomlyon a 2D square lattice L × L with the same densities. Asshown in FIG. 1 each pedestrian moves to the preferential di-rection with no backstep. The first (second) type of pedes-trian is eastbound (northbound), E pedestrian (N pedestrian)for short. For example, the E pedestrian could move to theeastward, northward and southward site while the N pedes-trian could move to the northward, eastward and westwardsite. The pedestrians exclude each other on a site. Thus, eachlattice site can be in one of three states: empty, occupied bythe E pedestrian, or occupied by the N pedestrian.Under periodic boundary conditions, the following stepsare repeated L times in one Monte Carlo step (MCS): (i)one site is selected randomly; (ii) if the selected site is empty,nothing happens; otherwise, if the selected site is occupied bythe E pedestrian, the nearest east, south and north neighbor-ing site are chosen as target site with probabilities q , (1 − q ) / − q ) /
2, respectively; otherwise, if the selected site isoccupied by the N pedestrian, the nearest north, east and westneighboring site are chosen as target site with probabilities q ,(1 − q ) / − q ) /
2; (iii) the pedestrian moves to the targetsite unless it is occupied.Under open boundary conditions, the E (N) pedestrians areinjected with probability α on the west (south) boundary andremoved with probability β on the other three boundaries. Atthe southwest corner, the E or N pedestrians are injected withprobability α/ III. RESULTSA. Periodic boundary
1. Simulation results
The lattice size is set as 100 ×
100 unless otherwise men-tioned. For each density, we simulate 100 runs. The result ofeach run is obtained after discarding the first 10 MCSs (astransient time) and averaged in the next 10 MCSs. q (1-q)/2 (1-q)/2 q(1-q)/2 (1-q)/2 FIG. 1: Illustration of the intersecting pedestrian flow at a crossing.The E and N pedestrians are represented by the solid and open cir-cles, respectively . q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 v density FIG. 2: The average velocity of each run against the pedestrian den-sity ρ as q = . , . , . , . , . × The average velocity v of each run against the pedestriandensity ρ for q = . , . , . , . , . v is defined as the average directed dis-tance divided by the MCS. When q =
1, our model reducesto the BML-R model [30]. One can see that two phases couldbe observed, i.e., the moving and intermediate phase. In themoving phase, all pedestrians can move. In the intermediatephase, the average velocity becomes a non-negligible smallvalue v > v =
0. There is a range of densitiesin which the two phases coexist and we denote the center ofthis range as ρ c . One can see that with the increase of q , thecritical density ρ c decreases.The three typical configurations for the intermediate phaseare shown in FIG. 3. Some pedestrians are stopped at theinterior of the cluster (stripe) while others could move alongthe border of the jamming stripes from the lower left to the (d)(c) (b)(a) FIG. 3: Three typical configurations of the intermediate phase.The parameters are L = 100, and (a) q =0.6, ρ =0.2,(b) q =0.6, ρ =0.7,(c) q =0.8, ρ =0.7,(d) q =0.6, ρ =0.9. The E pedestrian is indicated byblue and the N pedestrian is indicated by red. upper right corner. With the increase of density and q , thenumber of stripes in one row (column) increases (see FIG.3(b) and (c)).
2. Analytical result of moving phase
We have developed a mean field analysis for the averagevelocity in the moving phase by extending the method of ref-erence [30]. The E (N) pedestrian could move to the east, thenorth or the south (the north, the east or the west) site withdifferent probabilities. If we have selected a site occupied bythe E (N) type of pedestrian, then the probability that its east(north) site is empty is assumed to be p f , while the probabilitythat its north or south (west or east) site is empty is assumedto be p s .In order to calculate these p f and p s , 21 situations havebeen considered as shown in FIG. 4( a ). The dashed arrowsrepresents the target pedestrian. Since the symmetry of themodel, the target pedestrian is restricted to the E pedestrian.The solid arrows ↓ ( ← ) only represents the E (N) pedestrianmoving south (west). The dark and empty box represent apedestrian and an empty site, respectively.The left side of each subfigure is the existence probabilityof the corresponding situation at the current time. The rightside is the situation that the target pedestrian could move whenis chosen again. The moving probability of the target pedes-trian when is chosen again is shown in the subfigures G , H , I and J respectively.We explain the subfigures A A B D H and G in de-tail. Other situations could be obtained similarly. Subfigure A : On the left side of subfigure A q is the probability that * JII (1-q)(1/2)p s (1-q)(1/2)(1-p s )p s (1-q)(1/2)(1-p s )(1-p s )qp f ( ) F3F2F1D3D2D1 B3B2B1A3A2A1 JII JJJ HH GGH (1-q)(1/2)(1-p s ) current time is chosen the timechosen again q(1-p f )qp f (1-q)(1/2)(1-p s )( p f ) (1-q)(1/2)p s (1-q)(1/2)(1-p s )p f current time is chosen the timechosen again (1-q)(1/2)p s (1-q)(1/2)p s ( ) q(1-p f )p s qp f q(1-p f )(1-p s ) GB4 (1-q)(1/2)p s ( ) F4 I qp f ( ) E3E2E1 current time is chosen the timechosen again
HC3C2C1 HG (1-q)(1/2)(1-p s ) ( p f ) (1-q)(1/2)p s (1-q)(1/2)(1-p s )p f GC4 (1-q)(1/2)p s ( ) current time is chosen the timechosen again current time is chosen the timechosen againcurrent time is chosen the timechosen again H (a) J3J2J1H3H1H2 [(1/2)(1/2)qp f +(1/2)(1/2)(1-q)(1/2)p s ][1-( /2)q(1/2)-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][(1/2)(1/2)(1-q)(1/2)p s ][1-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][(1/2)(1/2)qp f +(1/2)(1/2)(1-q)(1/2)p s ][1-( /2)q(1/2)-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][1-( /2)q(1/2)-( /2)(1-q)(1/2)(1/2)] GH [(1/2)(1/2)(1-q)(1/2)p s ][1-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][(1/2)(1/2)qp f +(1/2)(1/2)(1-q)(1/2)p s ][1-( /2)(1-q)(1/2)(1/2)][1-( /2)q(1/2)-( /2)(1-q)(1/2)(1/2)][(1/2)(1/2)qp f +(1/2)(1/2)(1-q)(1/2)p s ][1-( /2)q(1/2)-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][1-( /2)(1-q)(1/2)(1/2)][1-( /2)q(1/2)-( /2)(1-q)(1/2)(1/2)] IJ The end ofcurrent time chosen again
The end ofcurrent time chosen again (b)
FIG. 4: The illustration of the mean field method. The dashed arrowsrepresents the E pedestrian. The solid arrows ↓ ( ← ) only representsthe E (N) pedestrian moving south (west). The dark, empty and graybox represent a pedestrian, an empty site and a site which is eitherempty or occupied, respectively. the target pedestrian moves east and (1 − p f ) is the probabilitythat the east site of d is occupied so that d could not movein the current time step. If the target pedestrian wants to moveto the east site successfully when it is chosen again, its eastsite must be empty. The probability that the east site of d becomes empty is shown in the subfigure H . Subfigure H : The left side of subfigure H is the configuration at the endof the current time of subfigure A
1. On the right site of H · · · ] corresponds to that of east, north-eastand east-east site, respectively.The first (1/2) in the first [ · · · ] is the probability that the eastsite of d is occupied by the E pedestrian, the second (1/2) isthe probability that the E pedestrian is chosen before d ischosen again, (1 − q )(1 /
2) is the probability that the E pedes-trian moves south and p s is the probability that the south-eastsite of d is empty. So that [(1 / / − q )(1 / p s ] is theprobability that the east site of d becomes empty first.The ( ρ/
2) in the second [ · · · ] is the probability that thenortheast site of d is occupied by the E pedestrian, (1 − q )(1 / /
2) is the probability that the E pedestrian in thenortheast site is chosen and moves south before d is chosenagain. Similarly, ( ρ/ − q )(1 / /
2) in the third [ · · · ] rep-resents that the east-east site is occupied by the N pedestrianand it moves west before d is chosen again.Thus [1 − ( ρ/ − q )(1 / / − ( ρ/ − q )(1 / / d . So the eastsite of d remains empty after the pedestrian in the east sitemoves out.The probabilities in subfigures H H H
2. Thus(1 / / qp f and (1 / / − q )(1 / p s in the first [ · · · ]corresponds to that of the N pedestrian and E pedestrian, re-spectively. Subfigure A : Similarly, on the left of subfigure A q is the same as thatof A p f is the probability that the east site of d is emptyso that d could move in the current time step, (1 − ρ ) is theprobability that the east-east site of d is empty. If the tar-get pedestrian wants to move to its east-east site successfullywhen it is chosen again, the east-east site must remain empty.The probability that the east-east site stays empty is shown in the subfigure G . Subfigure G : The left side of subfigure G is the configuration at the end ofcurrent time of subfigures A
3. On the right side of subfigure G , the three terms in [ · · · ] corresponds to that of east-east,north-east sites and south-east site, respectively.The ( ρ/
2) in the first [ · · · ] is the probability that theeast-east site of d is occupied by the N pedestrian. (1 − q )(1 / /
2) is the probability that the N pedestrian in theeast-east site is chosen and moves west before d is chosenagain. Similarly, the second [ · · · ] corresponds to that of north-east sites.Since the south-east site could be occupied by either theE pedestrian or the N pedestrian. Thus ( ρ/ q (1 /
2) and( ρ/ − q )(1 / /
2) in the third [ · · · ] corresponds to theN pedestrian and the E pedestrian, respectively.Thus [1-( ρ /2)(1 − q )(1/2)(1/2)][1-( ρ /2)(1 − q )(1/2)(1/2)][1-( ρ /2) q (1/2)-( ρ /2)(1 − q )(1/2)(1/2)] is the probability that noneof pedestrians moves to the east site of d before d is chosenagain. Subfigure B : On the left side of subfigure B
1, (1 − q )(1 /
2) is the probabil-ity that the E pedestrian moves north. (1 − p s ) and (1 − p f ) arethe probabilities that the north and east site of ↑ are occupied,respectively. So that ↑ could not move in the current time step.If the target pedestrian wants to move to the east site success-fully when it is chosen again, the east site must be empty. Theprobability that the east site becomes empty is also shown inthe subfigure H . Other subfigures:
The subfigures D ∼ D E ∼ E F ∼ F A ∼ A B ∼ B C ∼ C D
1, (1 − q )(1 /
2) isthe probability that the E pedestrian moves north and (1 − p s )is the probability that the north site of ↑ is occupied so that ↑ could not move north in the current time step. If the targetpedestrian wants to move to the north site successfully when itis chosen again, the north site must be empty. The probabilitythat the north site becomes empty is shown in the subfigure J .The meanings of the terms in J are similar to that of H .Since the E pedestrian can move east when it is chosenagain in the situation A ∼ A B ∼ B C ∼ C D ∼ D E ∼ E F ∼ F p f = [ q (1 − p f ) + qp f ρ +
12 (1 − q )(1 − p s )(1 − p f ) +
12 (1 − q ) p s ρ +
12 (1 − q )(1 − p s )(1 − p f ) +
12 (1 − q ) p s ρ ] { [ 12 12 (1 − q ) 12 p s ][1 − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ] + [ 12 12 qp f +
12 12 (1 − q ) 12 p s ][1 − ρ q − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ] + [ 12 12 qp f +
12 12 (1 − q ) 12 p s ][1 − ρ q − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ] } + [ qp f (1 − ρ ) +
12 (1 − q )(1 − p s ) p f +
12 (1 − q ) p s (1 − ρ ) +
12 (1 − q )(1 − p s ) p f +
12 (1 − q ) p s (1 − ρ )] { [1 − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ][1 − ρ q − ρ − q ) 12 12 ] } . (1) p s = [ 12 (1 − q )(1 − p s ) +
12 (1 − q ) p s ρ +
12 (1 − q )(1 − p s )(1 − p s ) + q (1 − p f )(1 − p s ) + qp f ρ ] { [ 12 12 qp f +
12 12 (1 − q ) 12 p s ][1 − ρ q − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ] + [ 12 12 qp f +
12 12 (1 − q ) 12 p s ][1 − ρ − q ) 12 12 ][1 − ρ q − ρ − q ) 12 12 ] + [ 12 12 (1 − q ) 12 p s ][1 − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ] } + [ 12 (1 − q ) p s (1 − ρ ) +
12 (1 − q )(1 − p s ) p s +
12 (1 − q ) p s + q (1 − p f ) p s + qp f (1 − ρ )] { [1 − ρ − q ) 12 12 ][1 − ρ − q ) 12 12 ][1 − ρ q − ρ − q ) 12 12 ] } . (2) simulation analytical < v > density (c) simulation analytical < v > density (d) < v > density simulation analytical (a) simulation analytical < v > density (b) FIG. 5: The comparison of the simulation results with the mean fieldanalysis for the moving phase. The dashed lines are analytical resultsand black squares are simulation ones. The average velocity < v > are the average of 100 runs for each density. (a) q =0.6, (b) q =0.7,(c) q =0.8, (d) q =0.9. The average velocity can be obtained from p f , < v > = qp f . (3)FIG. 5 compares the simulation results with the analyticalresults. One can see that the analytical results are in goodagreement with the simulation results, which proves that ouranalytical method is effective.
3. Analytical results of intermediate phase
We have developed two methods to analyse the average ve-locity v in the intermediate phase. As shown in FIG. 6, thesolid circle represents the E pedestrian, whose coordinates are( i , j ). Then its east, north and south neighbors’ coordinatesare ( i + , j ), ( i , j −
1) and ( i , j + j-1 i-1 Barrier (b)
Barrier (a) ij+1j i+1 (1-q)/2(1-q)/2 qq q (1-q)/2 (1+q)/2
FIG. 6: The 2D model in the intermediate phase is transformed intothe 1D model, which is similar to that of reference [32–34]. d < τ ( i , j ) > dt = { q < τ ( i − , j ) (1 − τ ( i , j ) ) > + (1 − q )(1 / < τ ( i , j − (1 − τ ( i , j ) ) > + (1 − q )(1 / < τ ( i , j + (1 − τ ( i , j ) ) > }−{ q < τ ( i , j ) (1 − τ ( i + , j ) ) > + (1 − q )(1 / < τ ( i , j ) (1 − τ ( i , j − ) > + (1 − q )(1 / < τ ( i , j ) (1 − τ ( i , j + ) > } = , (4)where τ ( i , j ) = i , j ). τ ( i , j ) = i , j ) is occupied while τ ( i , j ) = < · · · > represents an average with respect toall the microstates. The terms q < τ ( i − , j ) (1 − τ ( i , j ) ) > , (1 − q )(1 / < τ ( i , j − (1 − τ ( i , j ) ) > and (1 − q )(1 / < τ ( i , j + (1 − τ ( i , j ) ) > represent a flux into site ( i , j ). While q < τ ( i , j ) (1 − τ ( i + , j ) ) > , (1 − q )(1 / < τ ( i , j ) (1 − τ ( i , j − ) > and (1 − q )(1 / <τ ( i , j ) (1 − τ ( i , j + ) > represent a flux out of site ( i , j ).Since the jamming stripes are oriented along the diagonaldirection, < τ ( i , j − > ≈ < τ ( i − , j ) > and < τ ( i , j + > ≈ < τ ( i + , j ) > are assumed. Equation (4) becomes d < τ ( i , j ) > dt = (1 − q )(1 / { < τ ( i + , j ) > (1 − < τ ( i , j ) > ) − < τ ( i , j ) > (1 − < τ ( i − , j ) > ) } + (1 + q )(1 / { < τ ( i − , j ) > (1 − < τ ( i , j ) > ) − < τ ( i , j ) > (1 − < τ ( i + , j ) > ) } = . (5)where the approximation < τ i τ i + > = < τ i >< τ i + > is used.From equation (5) one can see that flux only occurs amongsites ( i , j ), ( i − , j ) and ( i + , j ), ie., the horizontal direc-tion. The two dimensional model (FIG. 6(a)) is reduced toone dimension model (FIG. 6(b)). The density distribution ofintermediate phase in one row (column) is similar to that ofreference[32–34].Therefore equation (6) can be obtained by solving equation(5), < τ i + > = < τ i > + q < τ i >< τ i − > < τ i > + − q , (6)in which the label j is taken off. The density of each site < τ i > could be obtained by the iteration of equation (6). Thedensities of the first and second site adopt < τ > = < τ > → v could be calculated by the following equation v = ∞ X i = q < τ i > (1 − < τ i + > ) . (7)There are n s × L
1D lattice for the E pedestrian and the Npedestrian respectively when the system has n s stripes. Theaverage velocity of the intermediate phase is v = n s Lv ( ρ/ L . (8)The second method uses the results of reference [32, 33] forthe 1D lattice in FIG. 6(b), p ( τ , · · · , τ i , · · · , τ L ) = Z Y i ( q ′ − q ′ ) i τ i = Z ( q ′ − q ′ ) P i = Li = i τ i = Z e ln ( q ′ − q ′ ) P i = Li = i τ i , (9)where p ( τ , · · · , τ i , · · · , τ L ) denotes the probability density ofmicrostates ( τ , · · · , τ i , · · · , τ L ) and Z is a normalization fac-tor, such that the sum of p ( τ , · · · , τ i , · · · , τ L ) over all allowedconfigurations is 1. Here , q ′ is the probability the pedes-trian moving to the right while 1 − q ′ to the left in the 1Dlattice. From FIG. 6(b), one can see that q ′ = (1 + q ) / − q ′ = (1 − q ) / p ( τ , · · · , τ i , · · · , τ L ) = Z e ln ( + q − q ) P i = Li = i τ i . (10)Since the exclusion property of particles, the density of eachsite satisfies the Fermi distribution < τ i > = e − ( i − u ) ln ( + q − q ) + , (11)where µ ≫
0. When ( i − µ ) in equation (11) is substituted by i , i.e. < τ i > = e − ( i ) ln (1 + q − q ) + ,then equation (7) becomes v = ∞ X i = −∞ q < τ i > (1 − < τ i + > ) . (12)Since ∞ X i = −∞ f ( i ) = ∞ X i = −∞ Z ∞−∞ f ( x ) δ ( x − i ) dx = Z ∞−∞ f ( x ) ∞ X i = −∞ δ ( x − i ) dx = Z ∞−∞ f ( x ) ∞ X m = −∞ e π imx dx = ∞ X m = −∞ Z ∞−∞ f ( x ) e π imx dx , (13)then v = ∞ X m = −∞ Z ∞−∞ e π imx q < τ ( x ) > (1 − < τ ( x + > ) dx . (14)Compared to the term m = m , v ≈ Z ∞−∞ q < τ ( x ) > (1 − < τ ( x + > ) dx ≈ − q . (15)Substituting equation (15) into equation(16), the averagevelocity of the intermediate phase is v = n s Lv ( ρ/ L = n s (1 − q ) ρ L . (16)FIG. 7 compares the simulation results with the mean fieldresults. One can see that the analytical results are in goodagreement with the simulation results. As shown in FIG. 3(d),when the density is in the high density region of the interme-diate phase, the jamming stripes meet and merge with eachother, which makes equation (16) not suitable. One can seethat with the increasing of q , the density where multi stripesbegin to appear decreases. v density simulation n s =1 n s =2 n s =3 (a) v density simulation n s =1 n s =2 n s =3 n s =4 (b) FIG. 7: The comparison of the simulation results with the mean fieldanalysis for the intermediate phase as q =(a) 0.6, (b) 0.8 for the latticesize 100 × v is the average velocity of each run.
4. The analytic result when the system only has one empty site
When the system has only one empty site, the model canbe seen as the random walk of the empty site on the 2D lat-tice. However, a self-organized pattern with stripes along thediagonals from the upper-right to the lower-left corners wasobserved (see FIG. 8(a)). FIG. 8(b) shows the moving trajec-tory of the empty site in 10 ,
000 MCS. One can see that theempty site moves along the interface of jamming stripe. Next,we analyse the average movement < v mov > of empty site inone MCS. As shown in FIG 9(a), according to the neighbors’states of empty site, there are 16 possible configurations, i.e. S a ∼ S a . The probabilities of configurations S a ∼ S a aredenoted as p ∼ p , respectively.We develop an analytical method to solve p ∼ p . Theevolution process of S a is shown in FIG. 10(a). One can seethat S a can be evolved from three configurations S c , S c and S c . The corresponding transition probabilities are q , (1 − q ) / − q ) /
2, respectively. While S a can also evolve into otherthree configurations S d , S d and S d . The corresponding tran-sition probabilities are q , (1 − q ) / − q ) /
2, respectively.Thus, the master equation for configurations S a is d p dt = [ p S c q + p S c (1 − q )(1 / + p S c (1 − q )(1 / − [ p q + p (1 − q )(1 / + p (1 − q )(1 / = , (17)where p S c , p S c and p S c are the probability of configurations S c , S c and S c , respectively.From FIG. 10(b) one can see that the configurations S c iscomposed of S b and S e . We assume that p S c = p S b p ( S c | S b ) ≈ p S b p S e , (18)where p S b and p S e are the probabilities of configurations S b and S e , respectively.As shown in FIG. 9(b), S b represents that the south neigh-bor of the empty site is the N pedestrian and the other three (b) (a) FIG. 8: (a) The typical configuration when the system only has oneempty site as q = . (b) S S S S S S S S S S S S S S S S S S S S S S S S (a) FIG. 9: (a) Sixteen typical configurations of the empty site’s neigh-bors. − ( | ) represents the E (N) pedestrian. (b) Eight typical con-figurations of the empty site’s neighbors. The solid box represents apedestrian which is either the E pedestrian or the N pedestrian. neighbors can be the E pedestrian or the N pedestrian. There-fore, the probability of S b is the sum of the probabilities ofthe eight typical configurations S a , S a , S a , S a , S a , S a , S a and S a , i.e., p S b = p + p + p + p + p + p + p + p . (19)These p S b ∼ p S b can be obtained similarly.Since the empty site moves diagonally along the interfaceof jamming stripe, we propose a method to estimate p S e . Weassume that the probabilities of the southwest (northeast) siteoccupied by the E pedestrian and the N pedestrian are both1 /
2. We assume the probability of the southeast, east-eastand south-south site occupied by the E pedestrian are both p RE while the probabilities for the N pedestrian are p RN . Similarly,the probabilities of the northwest, west-west and north-northsite occupied by the E pedestrian are p LE , while the probabili-ties for the N pedestrian are p LN . Thus, the probability of the S e is S S S S S S S S (b) S (a)= + S FIG. 10: (a) The evolution process of S a . (b) The configurations S c is composed of S b and S e . − ( | ) represents the E (N) pedestrian. p S e ≈ p RN p RN . (20)From FIG. 9(b) and FIG. 10(b), one can see that p RN approx-imately equals the probability of S b , i.e., p RN ≈ p S b . Similarly, p RE , p LN and p LE approximately equal p S b , p S b and p S b respec-tively. The four probabilities are shown in equation (21) p LE ≈ p S b = p + p + p + p + p + p + p + p ; p LN ≈ p S b = p + p + p + p + p + p + p + p ; p RE ≈ p S b = p + p + p + p + p + p + p + p ; p RN ≈ p S b = p + p + p + p + p + p + p + p . (21)Substituting equations (18)-(21) into (17), we have d p dt = [ 12 p r p r p S b q + p r p r p S b − q + p l p l p S b − q − [ p q + p − q + p − q = [ 12 ( p + p + p + p + p + p + p + p ) ( p + p + p + p + p + p + p + p ) q +
12 ( p + p + p + p + p + p + p + p ) − q +
12 ( p + p + p + p + p + p + p + p ) − q − [ p q + p − q + p − q = . (22)The equations for p ∼ p can be obtained similarly. How-ever, note that only fifteen of the sixteen equations are inde-pendent ones.The conservation of probability requires that X i = p i = . (23) Now, we have sixteen equations for the sixteen variables.We cannot obtain analytical result of the equations. Instead,we can obtain numerical result.The average movement of empty site < v mov > in one MCScan be calculated from p p < v mov > = p ( q + − q + p ( q + − q + p ( q + − q + p − q )2 + p (2 q + − q + p ( q + − q + p − q )2 + p − q )2 + p (2 q + − q + p q + p ( q + − q + p − q )2 + p ( q + − q + p ( q + − q + p (2 q + − q + p ( q + − q . (24)FIG. 11 compares the simulation results with the analytical results. One can see that the analytical results are in approxi- < v m o v > q simulation analytical FIG. 11: The average movement of the empty site in one MCS < v mov > against q . The solid and open squares are analytical andsimulation results respectively. Jamming PhaseMoving Phase q=0.5 q=0.7 q=0.9
FIG. 12: Phase diagram of the model under open boundary condi-tions for lattice size L =
100 . mate agreement with simulation ones. This is due to that thecorrelation among the neighbors of empty site. More effectivemethods are needed for the probability of S e , etc. B. Open boundary
FIG.12 shows the phase diagram under open boundary con-ditions when q = . , . , .
9. Similar to the result of refer-ence [30, 34], there are two phases, i.e. the moving phase andthe jamming phase.Three typical configurations for q = . c f q c f (a) c q c /f simulation (b) FIG. 13: (a)The simulation results of ρ c and f against q as α = . β = L = α c ≈ ρ c / f and α c of FIG.12 against q . (b)(a) (c) FIG. 14: Three typical configurations of the model with open bound-ary conditions at q = .
7. The parameters are L = β = . α = . = . α = .
2. The E pedestrian is indicatedby blue and the N pedestrian is indicated by red. f l o w q=0.7 den s i t y < v > f l o w < v > den s i t y q=0.9 FIG. 15: The global density, the average velocity and the flow in themoving phase at (a) q = . q = . β =
1. The simulationresults for the global density, average velocity < v > and flow are theaverage of 100 runs. from the result of reference [30], the shape of jamming regionbecomes diamond. Because the E (N) pedestrian can moveout from the up (right) boundary (see FIG. 14(b)).From FIG.12, one can see that the critical injection proba-bility α c increases firstly then decreases with the increase of q .The explanation is as follows. As shown in FIG.15, the den-sity increases almost linearly with α , i.e., ρ ≈ α ∗ f ( q ). When α increases to α c , the density exceeds ρ c of period bound-ary condition and the jam happens, i.e., α c ≈ ρ c ( q ) / f ( q ).FIG.13(a) shows the simulation results of ρ c and f against q . One can see that both ρ c and f decrease with the increaseof q . However the decline rates are different. The decline rateof f is greater than that of ρ c when q < .
8, while smallerwhen q > .
8. The α c ≈ ρ c / f shown in FIG.13(b) agreeswith α c of FIG.12 qualitatively.We study the global density ρ , the average velocity < v > and the flow J in the moving phase. Because of the conserva-tion of flow, the flow in the bulk equals the inflow; i.e. J = ρ qp f = α (1 − ρ ) . (25)Combining the equations (1)-(3) and (25), we obtain the ρ and < v > for each α and q . Then the flow can be calculated byusing equation (25). The analytical results are shown in FIG.15 and are in good agreement with the simulation results. IV. CONCLUSION
In this paper, the intersecting pedestrian flow on two-dimensional lattice has been studied. Under periodic bound-ary condition, the intermediate phase in which some pedes-trians could move along the border of jamming stripes wereobserved. The density where multi stripes begin to appeardecreases with the increase of q . We have developed a meanfield analysis for the moving phase by extending the method of[30]. The analytical results agree with the simulation resultswell. The average velocity of intermediate phase was obtainedby the analytical result of a 1D model, which agree with thesimulation results well. When the system has only one emptysite, the average movement in one MCS was obtained by meanfield analysis. There is a little deviation between the simula-tion and analytical result because of the correlation among thepedestrians. More accurate methods will be developed in thefuture work.Under the open boundary conditions, the moving phase andjamming phase were observed. The shape of jamming regionchanges into diamond. The critical injection probability α c shows nontrivially against q . The analytical results for theflow rate, average velocity and density in the moving phasewere obtained and agree with the simulation results well.The analytical methods in the paper could be generalized tothe parallel updating models. The model also can be extendedto other complex environment such as the subway, classroom. A. Acknowledgments
This work is funded by the National Natural ScienceFoundation of China (Grant Nos. 71671058, 71301042,71431003), the Doctoral Program of the Ministry of Educa-tion (No. 20130111120027), Fundamental Research Fundsfor the Central Universities Singapore Ministry of EducationAcademic Research Fund Tier 2 (Grant No. MOE2013-T2-2-033). The numerical calculations in this paper have been doneon the supercomputing system in the Supercomputing Centerof University of Science and Technology of China.
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